Alewyn P. Burger

Vertex Covers and Secure Domination in Graphs

Let G = (V,E) be a graph and let S ⊆ V. The set S is a dominating set of G if every vertex in V \ S is adjacent to some vertex in S. The set S is a secure dominating set of G if for each u ∈ V \ S, there exists a vertex v ∈ S such that uv ∈ E and (S \ {v}) ∪ {u} is a dominating set of G. The minimum cardinality of a secure dominating set in G is the secure domination number γ s(G) of G. We show that if G is a connected graph of order n with minimum degree at least two that is not a 5-cycle, then γ s(G) ≤ n/2 and this bound is sharp. Our proof uses a covering of a subset of V(G) by vertex-disjoint copies of subgraphs each of which is isomorphic to K 2 or to an odd cycle.

On the domination number of prisms of graphs

For a permutation π of the vertex set of a graph G, the graph πG is obtained from two disjoint copies G1 and G2 of G by joining each v in G1 to π(v) in G2. Hence if π = 1, then πG = K2 × G, the prism of G. Clearly, γ(G) ≤ γ(πG) ≤ 2γ(G). We study graphs for which γ(K2 × G) = 2γ(G), those for which γ(πG) = 2γ(G) for at least one permutation π of V (G) and those for which γ(πG) = 2γ(G) for each permutation π of V (G).

On minimum secure dominating sets of graphs

AbstractA subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X/{v}) ᑌ {u} is again a dominating set of G, in which case v is called a defender. The secure domination number of G is the cardinality of a smallest secure dominating set of G. In this paper, we show that every graph of minimum degree at least 2 possesses a minimum secure dominating set in which all vertices are defenders. We also characterise the classes of graphs that have secure domination numbers 1, 2 and 3.Abstract A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X/{v}) ᑌ {u} is again a dominating set of G, in which case v is called a defender. The secure domination number of G is the cardinality of a smallest secure dominating set of G. In this paper, we show that every graph of minimum degree at least 2 possesses a minimum secure dominating set in which all vertices are defenders. We also characterise the classes of graphs that have secure domination numbers 1, 2 and 3.

Edge criticality in secure graph domination

A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set X{v}{u} is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is q-critical if the smallest arbitrary subset of edges whose removal from G necessarily increases the secure domination number, has cardinality q. In this paper we characterise q-critical graphs for all admissible values of q and determine the exact values of q for which members of various infinite classes of graphs are q-critical.

A linear algorithm for secure domination in trees

Abstract A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X , there is a neighbouring vertex v of u in X such that the swap set ( X − { v } ) ∪ { u } is again a dominating set of G . The secure domination number of G , denoted by γ s ( G ) , is the cardinality of a smallest secure dominating set of G . A linear algorithm is proposed in this paper for finding a minimum secure dominating set and hence the value γ s ( T ) for a tree T . The algorithm is based on a strategy of repeatedly pruning away pendent spiders of T after having dominated them securely.

An Infinite Family of Planar Hypohamiltonian Oriented Graphs

Carsten Thomassen asked in 1976 whether there exists a planar hypohamiltonian oriented graph. We answer his question by presenting an infinite family of planar hypohamiltonian oriented graphs, the smallest of which has order 9. A computer search showed that 9 is the smallest possible order of a hypohamiltonian oriented graph.

Balanced minimum covers of a finite set

In this paper a method of enumeration for n-balanced, labelled, minimum covers of a finite set of m indistinguishable elements is developed. The method is then used to tabulate the number of such covers for small values of m and n by means of a generating function.

An asymptotic analysis of the evolutionary spatial prisoner's dilemma on a path

In this paper, we consider the Evolutionary Spatial Prisoners Dilemma (ESPD) in which players are modelled by the vertices of an underlying graphG representing some spatial organisational structure amongst the players. During each round of the ESPD every pair of adjacent players in G play a classical prisoners dilemma against each other, and they update their strategies from one round to the next based on the perceived success achieved by the strategies of neighbouring players during the previous round. In this way, players are able to adapt and learn from each others strategies as the game progresses without being able to rationalise good strategies. We characterise all steady states of the ESPD for the case where G is a path, and we also characterise the structures of those initial states that lead to the emergence of persistent substates of cooperation over time. We finally determine analytically (i.e. without using simulation) the probability that the games states will evolve from a randomly generated initial state towards a steady state which accommodates some form of persistent cooperation. More specifically, we show that there exists a range of game parameters for which the likelihood of the emergence of persistent cooperation increases to almost certainty as the length of the path increases.

Scheduling multi-colour print jobs with sequence-dependent setup times

In this paper, a scheduling problem is considered which arises in the packaging industry. Plastic and foil wrappers used for packaging candy bars, crisps and other snacks typically require overlay printing with multiple colours. Printing machines used for this purpose usually accommodate a small number of colours which are loaded into a magazine simultaneously. If two consecutively scheduled print jobs require significantly different colour overlays, then substantial down times are incurred during the transition from the former magazine colour configuration to the latter, because ink cartridges corresponding to colours not required for the latter job have to be cleaned after completion of the former job. The durations of these down times are therefore sequence dependent (the washing and refilling time is a function of the number of colours in which two consecutive printing jobs differ). It is consequently desirable to schedule print jobs so that the accumulated down times associated with all magazine colour transitions are as short as possible for each printing machine. We show that an instance of this scheduling problem can be modelled as the well-known tool switching problem, which is tractable for small instances only. The problem can, however, be solved rather effectively in heuristic fashion by decomposing it into two subproblems: a job grouping problem (which can be modelled as a unicost set covering problem) and a group sequencing problem (which is a generalisation of the celebrated travelling salesman problem). We solve the colour print scheduling problem both exactly and heuristically for small, randomly generated test problem instances, studying the trade-off between the time efficiency and solution quality of the two approaches. Finally, we apply both solution approaches to real problem data obtained from a printing company in the South African Western Cape as a special case study.

On the ?(d) - chromatic number of a complete balanced multipartite graph

In this paper we solve (approximately) the problem of finding the minimum number of colours with which the vertices of a complete, balanced, multipartite graph G may be coloured such that the maximum degrees of all colour class induced subgraphs are at most some specified natural number d. The minimum number of colours in such a colouring is referred to as the Delta(d)–chromatic number of G. The problem of finding the Delta(d)–chromatic number of a complete, balanced, multipartite graph has its roots in an open graph theoretic characterisation problem and has applications conforming to the generic scenario where users of a system are in conflict if they require access to some shared resource. These conflicts are represented by edges in a so–called resource access graph, where vertices represent the users. An efficient resource access schedule is an assignment of the users to a minimum number of groups (modelled by means of colour classes) where some threshold d of conflict may be tolerated in each group. If different colours are associated with different time periods in the schedule, then the minimum number of groupings in an optimal resource access schedule for the above set of users is given by the Delta(d)–chromatic number of the resource access graph. A complete balanced multipartite resource access graph represents a situation of maximum conflict between members of different user groups of the system, but where no conflict occurs between members of the same user group (perhaps due to an allocation of diverse duties to the group members).